 
Summary: A Counterexample to the Generalized LinialNisan Conjecture
Scott Aaronson #
Abstract
In earlier work [1], we gave an oracle separating the relational versions of BQP and the
polynomial hierarchy, and showed that an oracle separating the decision versions would follow
from what we called the Generalized LinialNisan (GLN) Conjecture: that ``almost kwise in
dependent'' distributions are indistinguishable from the uniform distribution by constantdepth
circuits. The original LinialNisan Conjecture was recently proved by Braverman [7]; we o#ered
a $200 prize for the generalized version. In this paper, we save ourselves $200 by showing that
the GLN Conjecture is false, at least for circuits of depth 3 and higher.
As a byproduct, our counterexample also implies that # p
2 ## P NP relative to a random oracle
with probability 1. It has been conjectured since the 1980s that PH is infinite relative to a
random oracle, but the highest levels of PH previously proved separate were NP and coNP.
Finally, our counterexample implies that the famous results of Linial, Mansour, and Nisan
[12], on the structure of AC 0 functions, cannot be improved in several interesting respects.
1 Introduction
Proving an oracle separation between BQP and PH is one of the central open problems of quantum
complexity theory. In a recent paper [1], we reported the following progress on the problem:
(1) We constructed an oracle relative to which FBQP ## FBPP PH , where FBQP and FBPP PH are
